Tail of a linear diffusion with Markov switching

Abstract

Let Y be an Ornstein-Uhlenbeck diffusion governed by a stationary and ergodic Markov jump process X: dYt = a(Xt)Yt dt + σ(Xt) dWt, Y0 = y0. Ergodicity conditions for Y have been obtained. Here we investigate the tail propriety of the stationary distribution of this model. A characterization of either heavy or light tail case is established. The method is based on a renewal theorem for systems of equations with distributions on R.1. Introduction. The discrete-time models Y = (Y n , n ∈ N) governed by a switching process X = (X n , n ∈ N) fit well to the situations where an autonomous process X is responsible for the dynamic (or regime) of Y . These models are parsimonious with regard to the number of parameters, and extend significantly the case of a single regime. Among them, the socalled Markov-switching ARMA models are popular in several application fields, for example, in econometric modeling [see Hamilton (1989Hamilton ( , 1990]. More recently, continuous-time versions of Markov-switching models have been proposed in Basak, Bisi and Ghosh (1996) and Guyon, Iovleff and Yao (2004), where ergodicity conditions are established. In this paper we investigate the tail property of the stationary distribution of this continuous-time process. One of the main results (Theorem 2) states that this model can provide heavy tails, which is one of the major features required in nonlinear time-series modeling. Note that heavy tails may also be obtained by using a Lévy-driven Ornstein-Uhlenbeck (O.U.) process (without Markov switching); see Barndorff-Nielsen andShephard (2001) andBrockwell (2001).The considered process Y , called diffusion with Markov switching, is constructed in two steps:

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